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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-1uplex | Structured version Visualization version GIF version |
Description: A monuple is a set if and only if its coordinates are sets. (Contributed by BJ, 6-Apr-2019.) |
Ref | Expression |
---|---|
bj-1uplex | ⊢ (⦅𝐴⦆ ∈ V ↔ 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-pr11val 36988 | . . 3 ⊢ pr1 ⦅𝐴⦆ = 𝐴 | |
2 | bj-pr1ex 36989 | . . 3 ⊢ (⦅𝐴⦆ ∈ V → pr1 ⦅𝐴⦆ ∈ V) | |
3 | 1, 2 | eqeltrrid 2844 | . 2 ⊢ (⦅𝐴⦆ ∈ V → 𝐴 ∈ V) |
4 | df-bj-1upl 36981 | . . 3 ⊢ ⦅𝐴⦆ = ({∅} × tag 𝐴) | |
5 | p0ex 5390 | . . . 4 ⊢ {∅} ∈ V | |
6 | bj-xtagex 36972 | . . . 4 ⊢ ({∅} ∈ V → (𝐴 ∈ V → ({∅} × tag 𝐴) ∈ V)) | |
7 | 5, 6 | ax-mp 5 | . . 3 ⊢ (𝐴 ∈ V → ({∅} × tag 𝐴) ∈ V) |
8 | 4, 7 | eqeltrid 2843 | . 2 ⊢ (𝐴 ∈ V → ⦅𝐴⦆ ∈ V) |
9 | 3, 8 | impbii 209 | 1 ⊢ (⦅𝐴⦆ ∈ V ↔ 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 {csn 4631 × cxp 5687 tag bj-ctag 36957 ⦅bj-c1upl 36980 pr1 bj-cpr1 36983 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-bj-sngl 36949 df-bj-tag 36958 df-bj-proj 36974 df-bj-1upl 36981 df-bj-pr1 36984 |
This theorem is referenced by: bj-2uplex 37005 |
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